Distribution of Vertices Required a High‐Degree Condition on Partitions of Graphs Under Degree Constraints

ABSTRACT LetGbe a graph, and letf 1 , f 2 : V ( G ) → { 0 , 1 , 2 , … }be functions. LetT 1 ( G )be the union of edges shared by two cycles of order at most four, and letT 0 ( G ) = V ( G ) \ T 1 ( G ). In this paper, we prove that if foru ∈ T h ( G )withh ∈ { 0 , 1 },d G ( u ) ≥ f 1 ( u ) + f 2 ( u ) − 1 + 2 handmin { f 1 ( u ) , f 2 ( u ) } ≥ 2 − 2 h, thenGcan be partitioned into two subgraphsG 1andG 2such thatd G i( u ) ≥ f i ( u )for eachu ∈ V ( G i ). The result is a generalization of some known results and gives a distribution of vertices required by a high‐degree condition on partitions of graphs under degree constraints.